Factoring x^4 + x^3 + 2x - 4 = 0 (cubic equ)

  • Thread starter jenettezone
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In summary, the student attempted to solve an equation involving a 4th degree polynomial, but was unsuccessful. After trying to solve the equation using the rational root theorem, they were successful in finding a rational root of the cubic.
  • #1
jenettezone
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Homework Statement


x^4 + x^3 + 2x - 4 = 0


Homework Equations


N/A


The Attempt at a Solution


x^4 + x^3 + 2x - 4 = 0
x(x^3 + x^2 +2) = 4

i don't know what to do with this. i tried to factor (x^3 + x^2 +2), but i don't know how. I also have a feeling that I am not doing this correctly and that there should be a zero instead of a 4 on the right hand side of the equal sign...
 
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  • #3
I don't think there is, in general, a good way to factor a 4th degree polynomial. You can try synthetic division, if you think you have one factor: (x-1).
 
  • #4
clamtrox said:
Rational root theorem sure seems like an overkill in this case.

So what you do is you say aha! x = 1 and x = -2

1 and -2 are between obvious root candidates pointed to by the rational root theorem - so you have just used it.

Besides, you have also just solved the question for the OP, which is exactly a thing that you should not do.
 
  • #5
Borek said:
1 and -2 are between obvious root candidates pointed to by the rational root theorem - so you have just used it.

Besides, you have also just solved the question for the OP, which is exactly a thing that you should not do.

Oops, my bad. Also, I most definitely did not use rational root theorem; I used guessing. Just because I guess something and there exists a theorem that says my guess is good, doesn't mean I know or in any way care about the theorem. :-) Still, obviously it's a nice thing to know -- I wasn't thinking at all when posting.
 
  • #6
clamtrox said:
I wasn't thinking at all when posting.

:smile: happens to everyone :wink:
 
  • #7
Borek said:
This is not cubic.

Perhaps http://en.wikipedia.org/wiki/Rational_root_theorem would help (especially as a4 = 1).
But the polynomial he gets after factoring out x-1 is a cubic. Perhaps that is what he was talking about.

And the rational root theorem works nicely to find a rational root of that cubic, leaving just a quadratic equation to be solved. (The quadratic has complex roots.)
 

Related to Factoring x^4 + x^3 + 2x - 4 = 0 (cubic equ)

1. What is factoring?

Factoring is the process of breaking down a polynomial equation into simpler terms. In this case, we are looking to break down the equation x^4 + x^3 + 2x - 4 = 0 into factors that can be multiplied together to give us the original equation.

2. How do I know if I need to factor an equation?

You may need to factor an equation if you are trying to solve for a variable and the equation is in a polynomial form. Factoring can help simplify the equation and make it easier to solve.

3. What is the degree of this equation (x^4 + x^3 + 2x - 4 = 0)?

The degree of an equation is the highest exponent in the equation. In this case, the degree is 4, as x^4 is the highest exponent.

4. How do I factor a cubic equation?

To factor a cubic equation, we can use the rational roots theorem to find potential roots of the equation. Then, we can use synthetic division or long division to divide the equation by these potential roots and see which ones result in a remainder of 0. These roots can then be used to write the equation in factored form.

5. Can I use the quadratic formula to factor this equation (x^4 + x^3 + 2x - 4 = 0)?

No, the quadratic formula can only be used to solve quadratic equations, which have a degree of 2. This equation has a degree of 4, so we need to use a different method for factoring.

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